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Theorem ltbtwnnq 8570
Description: There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltbtwnnq  |-  ( A 
<Q  B  <->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem ltbtwnnq
StepHypRef Expression
1 ltrelnq 8518 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4725 . . . 4  |-  ( A 
<Q  B  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
32simprd 451 . . 3  |-  ( A 
<Q  B  ->  B  e. 
Q. )
4 ltexnq 8567 . . . 4  |-  ( B  e.  Q.  ->  ( A  <Q  B  <->  E. y
( A  +Q  y
)  =  B ) )
5 eleq1 2318 . . . . . . . . . 10  |-  ( ( A  +Q  y )  =  B  ->  (
( A  +Q  y
)  e.  Q.  <->  B  e.  Q. ) )
65biimparc 475 . . . . . . . . 9  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( A  +Q  y )  e.  Q. )
7 addnqf 8540 . . . . . . . . . . 11  |-  +Q  :
( Q.  X.  Q. )
--> Q.
87fdmi 5332 . . . . . . . . . 10  |-  dom  +Q  =  ( Q.  X.  Q. )
9 0nnq 8516 . . . . . . . . . 10  |-  -.  (/)  e.  Q.
108, 9ndmovrcl 5940 . . . . . . . . 9  |-  ( ( A  +Q  y )  e.  Q.  ->  ( A  e.  Q.  /\  y  e.  Q. ) )
116, 10syl 17 . . . . . . . 8  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( A  e. 
Q.  /\  y  e.  Q. ) )
1211simprd 451 . . . . . . 7  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  y  e.  Q. )
13 nsmallnq 8569 . . . . . . . 8  |-  ( y  e.  Q.  ->  E. z 
z  <Q  y )
1411simpld 447 . . . . . . . . . . . 12  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  A  e.  Q. )
151brel 4725 . . . . . . . . . . . . 13  |-  ( z 
<Q  y  ->  ( z  e.  Q.  /\  y  e.  Q. ) )
1615simpld 447 . . . . . . . . . . . 12  |-  ( z 
<Q  y  ->  z  e. 
Q. )
17 ltaddnq 8566 . . . . . . . . . . . 12  |-  ( ( A  e.  Q.  /\  z  e.  Q. )  ->  A  <Q  ( A  +Q  z ) )
1814, 16, 17syl2an 465 . . . . . . . . . . 11  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  A  <Q  ( A  +Q  z ) )
19 ltanq 8563 . . . . . . . . . . . . . 14  |-  ( A  e.  Q.  ->  (
z  <Q  y  <->  ( A  +Q  z )  <Q  ( A  +Q  y ) ) )
2019biimpa 472 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  z  <Q  y )  -> 
( A  +Q  z
)  <Q  ( A  +Q  y ) )
2114, 20sylan 459 . . . . . . . . . . . 12  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  z )  <Q  ( A  +Q  y ) )
22 simplr 734 . . . . . . . . . . . 12  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  y )  =  B )
2321, 22breqtrd 4021 . . . . . . . . . . 11  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  z )  <Q  B )
24 ovex 5817 . . . . . . . . . . . 12  |-  ( A  +Q  z )  e. 
_V
25 breq2 4001 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +Q  z )  ->  ( A  <Q  x  <->  A  <Q  ( A  +Q  z ) ) )
26 breq1 4000 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +Q  z )  ->  (
x  <Q  B  <->  ( A  +Q  z )  <Q  B ) )
2725, 26anbi12d 694 . . . . . . . . . . . 12  |-  ( x  =  ( A  +Q  z )  ->  (
( A  <Q  x  /\  x  <Q  B )  <-> 
( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z
)  <Q  B ) ) )
2824, 27cla4ev 2850 . . . . . . . . . . 11  |-  ( ( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z )  <Q  B )  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
2918, 23, 28syl2anc 645 . . . . . . . . . 10  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
3029ex 425 . . . . . . . . 9  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( z  <Q 
y  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
3130exlimdv 1933 . . . . . . . 8  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( E. z 
z  <Q  y  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
3213, 31syl5 30 . . . . . . 7  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( y  e. 
Q.  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) ) )
3312, 32mpd 16 . . . . . 6  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
3433ex 425 . . . . 5  |-  ( B  e.  Q.  ->  (
( A  +Q  y
)  =  B  ->  E. x ( A  <Q  x  /\  x  <Q  B ) ) )
3534exlimdv 1933 . . . 4  |-  ( B  e.  Q.  ->  ( E. y ( A  +Q  y )  =  B  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) ) )
364, 35sylbid 208 . . 3  |-  ( B  e.  Q.  ->  ( A  <Q  B  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
373, 36mpcom 34 . 2  |-  ( A 
<Q  B  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
38 ltsonq 8561 . . . 4  |-  <Q  Or  Q.
3938, 1sotri 5058 . . 3  |-  ( ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
4039exlimiv 2024 . 2  |-  ( E. x ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
4137, 40impbii 182 1  |-  ( A 
<Q  B  <->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   class class class wbr 3997    X. cxp 4659  (class class class)co 5792   Q.cnq 8442    +Q cplq 8445    <Q cltq 8448
This theorem is referenced by:  nqpr  8606  reclem2pr  8640
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-omul 6452  df-er 6628  df-ni 8464  df-pli 8465  df-mi 8466  df-lti 8467  df-plpq 8500  df-mpq 8501  df-ltpq 8502  df-enq 8503  df-nq 8504  df-erq 8505  df-plq 8506  df-mq 8507  df-1nq 8508  df-rq 8509  df-ltnq 8510
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